# Understanding Maturity Structure of Interest Rates | CFA Level I Fixed Income

Welcome to another lesson on Fixed Income! Today, we’ll dive into the maturity structure of interest rates, explore yield curves, learn about their types, and discover how to calculate forward rates and price bonds using them.

## Term Structure of Interest Rates and Yield Curves

The term structure of interest rates, also known as the maturity structure, refers to the relationship between interest rates or bond yields and their respective maturities. The yield curve plays a crucial role in understanding this relationship.

When dealing with yield curves, it’s essential to know which yield is being represented. In this lesson, we’ll discuss three types of yield curves using US Treasuries as an example:

### Coupon Bonds Yield Curve

A coupon bond yield curve shows the yield-to-maturities for coupon bonds at various maturities. Yields for bonds with maturities between data points are estimated by linear interpolation.

When constructing a yield curve, an analyst should make sure that the currency, credit risk, periodicity, liquidity, and coupon rates of the bonds are similar. However, coupon bond yields have a significant constraint: the varying coupon rates introduce different coupon reinvestment risks for each bond.

### Spot Rate Yield Curve

A spot rate yield curve is based on zero-coupon bonds, which eliminates the problem of varying coupon rates. An ideal dataset for a spot curve would be yields-to-maturity on a series of zero-coupon government bonds for a full range of maturities.

For US Treasuries, we can use stripped bonds as a reference. Stripped bonds have their coupon payments and principal repayment separated and sold individually. The yields-to-maturity of stripped bonds can be considered as yields of zero-coupon bonds and are used to produce the spot curve.

The US Treasuries spot curve is also known as the stripped curve or zero curve. Often, these government spot rates are interpreted as “risk-free” rates.

### Par Bond Yield Curve

A par bond yield curve, or par curve, is constructed from the spot curve. The yields on this curve reflect the coupon rate a hypothetical bond at each maturity would need to have to be priced at par.

To derive any point on the par curve from the spot curve, we can price a bond using spot rates. We discount all future cash flows using spot rates, and in this case, our unknown variable is the coupon rate, which is the payment.

The par curve has various applications, such as its relationship with forward rates.

## Forward Rates

A forward rate is a borrowing or lending rate for a loan to be made at some future date. It consists of two components: the date of commencement of the loan and the length or tenor of the loan.

Forward rates are affected by spot rates. To understand this relationship, consider the following example:

EXAMPLE

Imagine you need to take a 2-year loan. You have two options:
Option A: Take a 2-year loan
Option B: Take a 1-year loan, and then take another 1-year loan in the second year.

The forward rate is the interest rate that will make these two options equivalent. To calculate the forward rate, we can use the following formula:

Formula: (1 + S2)^2 = (1 + S1) * (1 + F1)

Where:
S1 = 1-year spot rate S2 = 2-year spot rate F1 = 1-year forward rate in one year
By solving this equation, we can find the 1-year forward rate in one year (F1).

## Price Bonds Using Yield Curves

To price a bond using the yield curve, we must first understand the bond’s cash flows. After determining the cash flows, we can use either the coupon bond yield curve, the spot rate curve, or the par bond yield curve to discount them.

In general, the spot rate curve is the most accurate method for pricing a bond because it accounts for the reinvestment risk of the coupons. However, using the spot rate curve requires more effort and data.

The coupon bond yield curve is more straightforward to use, but it may lead to inaccuracies due to varying coupon rates and reinvestment risks. The par bond yield curve, on the other hand, can be useful for pricing bonds at par.

## Conclusion

In this lesson, we covered the maturity structure of interest rates and yield curves, focusing on coupon bond yield curves, spot rate yield curves, and par bond yield curves. We also learned how to calculate forward rates and price bonds using yield curves. Understanding the relationship between interest rates and maturities is essential for fixed income professionals to manage risks, make investment decisions, and value bonds.

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